[Weinberg-Witten theorem](http://en.wikipedia.org/wiki/Weinberg%E2%80%93Witten_theorem) states
that massless particles (either composite or elementary) with spin j>1/2 cannot carry a Lorentz-covariant current, while massless particles with spin j>1 cannot carry a Lorentz-covariant stress-energy. The theorem is usually interpreted to mean that the graviton ( j=2 ) cannot be a composite particle in a relativistic quantum field theory.
Before I read its proof, I've not been able to understand this result. Because I can directly come up a counterexample, massless spin-2 field have a Lorentz covariant stress-energy tensor. For example the Lagrangian of massless spin-2 is massless Fierz-Pauli action:
S=∫d4x(−12∂ahbc∂ahbc+∂ahbc∂bhac−∂ahab∂bh+12∂ah∂ah)
We can calculate its energy-stress tensor by Tab=−2√−gδSδgab, so we get
Tab=−12∂ahcd∂bhcd+∂ahcd∂chdb−12∂ah∂chbc−12∂ch∂ahbc+12∂ah∂bh+ηabL
which is obviously a non-zero Lorentz covariant stress-energy tensor.
And for U(1) massless spin-1 field, we can also have the energy-stress tensor Tab=FacFb c−14ηabFcdFcd
so we can construct a Lorentz covariant current
Ja=∫d3xTa0 which is a Lorentz covariant current.
Therefore above two examples are seeming counterexamples of this theorem. I believe this theorem must be correct and I want to know why my above argument is wrong.